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To my mind, the problem with “percentile” is that it is used with several meanings – a) a point or line b) a range c) something else.
a) A point or line. This is the easiest to understand.
If a mother takes a child to the doctor worried that the child isn’t growing normally, the doctor will plot the child’s height and weight on a percentile chart. If he finds that the child is on the 56th percentile line for height and the 60th for weight, for his age, he can reassure the mother that the child is 'normal' - a little above average.
b) A range. This should also be easy if the location of the range is given.
“In the bottom 25th percentile” means somewhere between the 0 line and the 25th percentile line (as defined above). “In the top 25th percentile means somewhere between the 75th percentile line and the 100th percentile line.
The child from a) is in the middle 20th percentile range for both height and weight. Of course, he is also within the middle 50th, 75th and 99th percentile range, but he is not within the middle 10th percentile range.
c) This is where it gets difficult. The actual definition of percentile (or one of them) is the range below the percentile line at which a value occurs. So in the original post, being “in the 21st percentile” actually means being in the range below the 21st percentile, while “in the top 25th percentile” is actually a misuse of the term according to this definition, and should read “above the 75th percentile (line)”.
If the word is used clearly, it is almost indispensable (as 2006 implies).